LMFDB - Embedded newform 1682.2.b.i.1681.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1682,2,Mod(1681,1682)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1682.1681"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1682.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4308376200$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 30x^{10} + 341x^{8} + 1897x^{6} + 5456x^{4} + 7680x^{2} + 4096 $ x^12 + 30x^10 + 341x^8 + 1897x^6 + 5456x^4 + 7680*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.1
Root			$-3.41744i$ of defining polynomial
Character	$\chi$	$=$	1682.1681
Dual form			1682.2.b.i.1681.12

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -3.41744i q^{3} -1.00000 q^{4} -1.61551 q^{5} -3.41744 q^{6} -1.52091 q^{7} +1.00000i q^{8} -8.67893 q^{9} +1.61551i q^{10} -1.89654i q^{11} +3.41744i q^{12} +3.18562 q^{13} +1.52091i q^{14} +5.52091i q^{15} +1.00000 q^{16} +1.69929i q^{17} +8.67893i q^{18} +3.67893i q^{19} +1.61551 q^{20} +5.19761i q^{21} -1.89654 q^{22} -2.69101 q^{23} +3.41744 q^{24} -2.39014 q^{25} -3.18562i q^{26} +19.4074i q^{27} +1.52091 q^{28} +5.52091 q^{30} +1.78239i q^{31} -1.00000i q^{32} -6.48132 q^{33} +1.69929 q^{34} +2.45703 q^{35} +8.67893 q^{36} -7.07024i q^{37} +3.67893 q^{38} -10.8867i q^{39} -1.61551i q^{40} -5.03259i q^{41} +5.19761 q^{42} +1.89654i q^{43} +1.89654i q^{44} +14.0209 q^{45} +2.69101i q^{46} +2.57063i q^{47} -3.41744i q^{48} -4.68684 q^{49} +2.39014i q^{50} +5.80722 q^{51} -3.18562 q^{52} +8.24904 q^{53} +19.4074 q^{54} +3.06387i q^{55} -1.52091i q^{56} +12.5725 q^{57} +2.95027 q^{59} -5.52091i q^{60} +7.74593i q^{61} +1.78239 q^{62} +13.1998 q^{63} -1.00000 q^{64} -5.14639 q^{65} +6.48132i q^{66} -2.47909 q^{67} -1.69929i q^{68} +9.19638i q^{69} -2.45703i q^{70} -6.06870 q^{71} -8.67893i q^{72} +3.74813i q^{73} -7.07024 q^{74} +8.16816i q^{75} -3.67893i q^{76} +2.88446i q^{77} -10.8867 q^{78} -9.22066i q^{79} -1.61551 q^{80} +40.2870 q^{81} -5.03259 q^{82} -5.52313 q^{83} -5.19761i q^{84} -2.74521i q^{85} +1.89654 q^{86} +1.89654 q^{88} -15.9704i q^{89} -14.0209i q^{90} -4.84503 q^{91} +2.69101 q^{92} +6.09122 q^{93} +2.57063 q^{94} -5.94334i q^{95} -3.41744 q^{96} +14.9078i q^{97} +4.68684i q^{98} +16.4599i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} - 4 q^{6} - 6 q^{7} - 24 q^{9} + 6 q^{13} + 12 q^{16} + 2 q^{22} - 28 q^{23} + 4 q^{24} + 8 q^{25} + 6 q^{28} + 54 q^{30} - 40 q^{33} + 12 q^{34} + 18 q^{35} + 24 q^{36} - 36 q^{38} + 20 q^{42}+ \cdots - 4 q^{96}+O(q^{100}) $ 12 * q - 12 * q^4 - 4 * q^6 - 6 * q^7 - 24 * q^9 + 6 * q^13 + 12 * q^16 + 2 * q^22 - 28 * q^23 + 4 * q^24 + 8 * q^25 + 6 * q^28 + 54 * q^30 - 40 * q^33 + 12 * q^34 + 18 * q^35 + 24 * q^36 - 36 * q^38 + 20 * q^42 + 36 * q^45 + 2 * q^49 + 22 * q^51 - 6 * q^52 + 6 * q^53 + 46 * q^54 + 38 * q^57 + 38 * q^59 - 34 * q^62 + 66 * q^63 - 12 * q^64 + 22 * q^65 - 42 * q^67 + 18 * q^71 + 24 * q^74 + 20 * q^78 + 4 * q^81 - 30 * q^82 - 4 * q^83 - 2 * q^86 - 2 * q^88 + 44 * q^91 + 28 * q^92 - 2 * q^93 + 16 * q^94 - 4 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times$.

$n$	$843$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 3.41744i	− 1.97306i	−0.163572	−	0.986531i	$-0.552302\pi$
$3$	− 3.41744i	− 1.97306i	0.163572	−	0.986531i	$-0.447698\pi$
$4$	−1.00000	−0.500000
$5$	−1.61551	−0.722477	−0.361238	−	0.932474i	$-0.617646\pi$
$5$	−1.61551	−0.722477	−0.361238	+	0.932474i	$0.617646\pi$
$6$	−3.41744	−1.39517
$7$	−1.52091	−0.574848	−0.287424	−	0.957803i	$-0.592799\pi$
$7$	−1.52091	−0.574848	−0.287424	+	0.957803i	$0.592799\pi$
$8$	1.00000i	0.353553i
$9$	−8.67893	−2.89298
$10$	1.61551i	0.510868i
$11$	− 1.89654i	− 0.571828i	−0.958255	−	0.285914i	$-0.907703\pi$
$11$	− 1.89654i	− 0.571828i	0.958255	−	0.285914i	$-0.0922972\pi$
$12$	3.41744i	0.986531i
$13$	3.18562	0.883532	0.441766	−	0.897130i	$-0.354352\pi$
$13$	3.18562	0.883532	0.441766	+	0.897130i	$0.354352\pi$
$14$	1.52091i	0.406479i
$15$	5.52091i	1.42549i
$16$	1.00000	0.250000
$17$	1.69929i	0.412138i	0.978537	+	0.206069i	$0.0660671\pi$
$17$	1.69929i	0.412138i	−0.978537	+	0.206069i	$0.933933\pi$
$18$	8.67893i	2.04564i
$19$	3.67893i	0.844004i	0.906595	+	0.422002i	$0.138673\pi$
$19$	3.67893i	0.844004i	−0.906595	+	0.422002i	$0.861327\pi$
$20$	1.61551	0.361238
$21$	5.19761i	1.13421i
$22$	−1.89654	−0.404343
$23$	−2.69101	−0.561114	−0.280557	−	0.959837i	$-0.590519\pi$
$23$	−2.69101	−0.561114	−0.280557	+	0.959837i	$0.590519\pi$
$24$	3.41744	0.697583
$25$	−2.39014	−0.478027
$26$	− 3.18562i	− 0.624751i
$27$	19.4074i	3.73496i
$28$	1.52091	0.287424
$29$	0	0
$29$	0	0
$30$	5.52091	1.00797
$31$	1.78239i	0.320127i	0.987107	+	0.160063i	$0.0511698\pi$
$31$	1.78239i	0.320127i	−0.987107	+	0.160063i	$0.948830\pi$
$32$	− 1.00000i	− 0.176777i
$33$	−6.48132	−1.12825
$34$	1.69929	0.291425
$35$	2.45703	0.415315
$36$	8.67893	1.44649
$37$	− 7.07024i	− 1.16234i	−0.813782	−	0.581170i	$-0.802595\pi$
$37$	− 7.07024i	− 1.16234i	0.813782	−	0.581170i	$-0.197405\pi$
$38$	3.67893	0.596801
$39$	− 10.8867i	− 1.74326i
$40$	− 1.61551i	− 0.255434i
$41$	− 5.03259i	− 0.785959i	−0.919547	−	0.392979i	$-0.871444\pi$
$41$	− 5.03259i	− 0.785959i	0.919547	−	0.392979i	$-0.128556\pi$
$42$	5.19761	0.802009
$43$	1.89654i	0.289219i	0.989489	+	0.144610i	$0.0461927\pi$
$43$	1.89654i	0.289219i	−0.989489	+	0.144610i	$0.953807\pi$
$44$	1.89654i	0.285914i
$45$	14.0209	2.09011
$46$	2.69101i	0.396768i
$47$	2.57063i	0.374966i	0.982268	+	0.187483i	$0.0600329\pi$
$47$	2.57063i	0.374966i	−0.982268	+	0.187483i	$0.939967\pi$
$48$	− 3.41744i	− 0.493266i
$49$	−4.68684	−0.669549
$50$	2.39014i	0.338016i
$51$	5.80722	0.813174
$52$	−3.18562	−0.441766
$53$	8.24904	1.13309	0.566546	−	0.824030i	$-0.308279\pi$
$53$	8.24904	1.13309	0.566546	+	0.824030i	$0.308279\pi$
$54$	19.4074	2.64102
$55$	3.06387i	0.413132i
$56$	− 1.52091i	− 0.203240i
$57$	12.5725	1.66527
$58$	0	0
$59$	2.95027	0.384093	0.192046	−	0.981386i	$-0.438488\pi$
$59$	2.95027	0.384093	0.192046	+	0.981386i	$0.438488\pi$
$60$	− 5.52091i	− 0.712746i
$61$	7.74593i	0.991765i	0.868390	+	0.495883i	$0.165155\pi$
$61$	7.74593i	0.991765i	−0.868390	+	0.495883i	$0.834845\pi$
$62$	1.78239	0.226364
$63$	13.1998	1.66302
$64$	−1.00000	−0.125000
$65$	−5.14639	−0.638331
$66$	6.48132i	0.797795i
$67$	−2.47909	−0.302870	−0.151435	−	0.988467i	$-0.548389\pi$
$67$	−2.47909	−0.302870	−0.151435	+	0.988467i	$0.548389\pi$
$68$	− 1.69929i	− 0.206069i
$69$	9.19638i	1.10711i
$70$	− 2.45703i	− 0.293672i
$71$	−6.06870	−0.720223	−0.360111	−	0.932909i	$-0.617261\pi$
$71$	−6.06870	−0.720223	−0.360111	+	0.932909i	$0.617261\pi$
$72$	− 8.67893i	− 1.02282i
$73$	3.74813i	0.438686i	0.975648	+	0.219343i	$0.0703913\pi$
$73$	3.74813i	0.438686i	−0.975648	+	0.219343i	$0.929609\pi$
$74$	−7.07024	−0.821898
$75$	8.16816i	0.943178i
$76$	− 3.67893i	− 0.422002i
$77$	2.88446i	0.328714i
$78$	−10.8867	−1.23267
$79$	− 9.22066i	− 1.03741i	−0.854955	−	0.518703i	$-0.826415\pi$
$79$	− 9.22066i	− 1.03741i	0.854955	−	0.518703i	$-0.173585\pi$
$80$	−1.61551	−0.180619
$81$	40.2870	4.47634
$82$	−5.03259	−0.555757
$83$	−5.52313	−0.606242	−0.303121	−	0.952952i	$-0.598029\pi$
$83$	−5.52313	−0.606242	−0.303121	+	0.952952i	$0.598029\pi$
$84$	− 5.19761i	− 0.567106i
$85$	− 2.74521i	− 0.297760i
$86$	1.89654	0.204509
$87$	0	0
$88$	1.89654	0.202172
$89$	− 15.9704i	− 1.69286i	−0.532498	−	0.846431i	$-0.678747\pi$
$89$	− 15.9704i	− 1.69286i	0.532498	−	0.846431i	$-0.321253\pi$
$90$	− 14.0209i	− 1.47793i
$91$	−4.84503	−0.507897
$92$	2.69101	0.280557
$93$	6.09122	0.631630
$94$	2.57063	0.265141
$95$	− 5.94334i	− 0.609773i
$96$	−3.41744	−0.348791
$97$	14.9078i	1.51366i	0.653612	+	0.756830i	$0.273253\pi$
$97$	14.9078i	1.51366i	−0.653612	+	0.756830i	$0.726747\pi$
$98$	4.68684i	0.473443i
$99$	16.4599i	1.65428i
$100$	2.39014	0.239014
$101$	− 0.183919i	− 0.0183006i	−0.999958	−	0.00915031i	$-0.997087\pi$
$101$	− 0.183919i	− 0.0183006i	0.999958	−	0.00915031i	$-0.00291267\pi$
$102$	− 5.80722i	− 0.575001i
$103$	−15.0468	−1.48261	−0.741303	−	0.671171i	$-0.765792\pi$
$103$	−15.0468	−1.48261	−0.741303	+	0.671171i	$0.765792\pi$
$104$	3.18562i	0.312376i
$105$	− 8.39678i	− 0.819442i
$106$	− 8.24904i	− 0.801218i
$107$	−0.304005	−0.0293893	−0.0146947	−	0.999892i	$-0.504678\pi$
$107$	−0.304005	−0.0293893	−0.0146947	+	0.999892i	$0.504678\pi$
$108$	− 19.4074i	− 1.86748i
$109$	−9.02387	−0.864330	−0.432165	−	0.901795i	$-0.642250\pi$
$109$	−9.02387	−0.864330	−0.432165	+	0.901795i	$0.642250\pi$
$110$	3.06387	0.292129
$111$	−24.1621	−2.29337
$112$	−1.52091	−0.143712
$113$	8.09737i	0.761737i	0.924629	+	0.380868i	$0.124375\pi$
$113$	8.09737i	0.761737i	−0.924629	+	0.380868i	$0.875625\pi$
$114$	− 12.5725i	− 1.17753i
$115$	4.34735	0.405392
$116$	0	0
$117$	−27.6478	−2.55604
$118$	− 2.95027i	− 0.271595i
$119$	− 2.58446i	− 0.236917i
$120$	−5.52091	−0.503987
$121$	7.40314	0.673013
$122$	7.74593	0.701284
$123$	−17.1986	−1.55075
$124$	− 1.78239i	− 0.160063i
$125$	11.9388	1.06784
$126$	− 13.1998i	− 1.17593i
$127$	9.51489i	0.844310i	0.906524	+	0.422155i	$0.138726\pi$
$127$	9.51489i	0.844310i	−0.906524	+	0.422155i	$0.861274\pi$
$128$	1.00000i	0.0883883i
$129$	6.48132	0.570648
$130$	5.14639i	0.451368i
$131$	14.2546i	1.24543i	0.782450	+	0.622713i	$0.213970\pi$
$131$	14.2546i	1.24543i	−0.782450	+	0.622713i	$0.786030\pi$
$132$	6.48132	0.564126
$133$	− 5.59530i	− 0.485174i
$134$	2.47909i	0.214161i
$135$	− 31.3528i	− 2.69842i
$136$	−1.69929	−0.145713
$137$	− 6.46454i	− 0.552303i	−0.961114	−	0.276151i	$-0.910941\pi$
$137$	− 6.46454i	− 0.552303i	0.961114	−	0.276151i	$-0.0890591\pi$
$138$	9.19638	0.782848
$139$	−14.2573	−1.20929	−0.604645	−	0.796495i	$-0.706685\pi$
$139$	−14.2573	−1.20929	−0.604645	+	0.796495i	$0.706685\pi$
$140$	−2.45703	−0.207657
$141$	8.78500	0.739831
$142$	6.06870i	0.509274i
$143$	− 6.04165i	− 0.505228i
$144$	−8.67893	−0.723244
$145$	0	0
$146$	3.74813	0.310198
$147$	16.0170i	1.32106i
$148$	7.07024i	0.581170i
$149$	8.23917	0.674979	0.337490	−	0.941329i	$-0.390422\pi$
$149$	8.23917	0.674979	0.337490	+	0.941329i	$0.390422\pi$
$150$	8.16816	0.666928
$151$	22.0029	1.79057	0.895285	−	0.445494i	$-0.146972\pi$
$151$	22.0029	1.79057	0.895285	+	0.445494i	$0.146972\pi$
$152$	−3.67893	−0.298401
$153$	− 14.7480i	− 1.19230i
$154$	2.88446	0.232436
$155$	− 2.87946i	− 0.231284i
$156$	10.8867i	0.871632i
$157$	0.135585i	0.0108208i	0.999985	+	0.00541041i	$0.00172220\pi$
$157$	0.135585i	0.0108208i	−0.999985	+	0.00541041i	$0.998278\pi$
$158$	−9.22066	−0.733556
$159$	− 28.1906i	− 2.23566i
$160$	1.61551i	0.127717i
$161$	4.09277	0.322556
$162$	− 40.2870i	− 3.16525i
$163$	10.2536i	0.803121i	0.915832	+	0.401561i	$0.131532\pi$
$163$	10.2536i	0.803121i	−0.915832	+	0.401561i	$0.868468\pi$
$164$	5.03259i	0.392979i
$165$	10.4706	0.815136
$166$	5.52313i	0.428678i
$167$	−16.7882	−1.29911	−0.649555	−	0.760314i	$-0.725045\pi$
$167$	−16.7882	−1.29911	−0.649555	+	0.760314i	$0.725045\pi$
$168$	−5.19761	−0.401004
$169$	−2.85183	−0.219371
$170$	−2.74521	−0.210548
$171$	− 31.9292i	− 2.44168i
$172$	− 1.89654i	− 0.144610i
$173$	−4.15473	−0.315878	−0.157939	−	0.987449i	$-0.550485\pi$
$173$	−4.15473	−0.315878	−0.157939	+	0.987449i	$0.550485\pi$
$174$	0	0
$175$	3.63517	0.274793
$176$	− 1.89654i	− 0.142957i
$177$	− 10.0824i	− 0.757839i
$178$	−15.9704	−1.19703
$179$	−10.6004	−0.792308	−0.396154	−	0.918184i	$-0.629655\pi$
$179$	−10.6004	−0.792308	−0.396154	+	0.918184i	$0.629655\pi$
$180$	−14.0209	−1.04505
$181$	21.7666	1.61790	0.808949	−	0.587878i	$-0.200037\pi$
$181$	21.7666	1.61790	0.808949	+	0.587878i	$0.200037\pi$
$182$	4.84503i	0.359137i
$183$	26.4713	1.95681
$184$	− 2.69101i	− 0.198384i
$185$	11.4220i	0.839763i
$186$	− 6.09122i	− 0.446630i
$187$	3.22276	0.235672
$188$	− 2.57063i	− 0.187483i
$189$	− 29.5169i	− 2.14704i
$190$	−5.94334	−0.431175
$191$	14.5209i	1.05069i	0.850888	+	0.525346i	$0.176064\pi$
$191$	14.5209i	1.05069i	−0.850888	+	0.525346i	$0.823936\pi$
$192$	3.41744i	0.246633i
$193$	4.80523i	0.345888i	0.984932	+	0.172944i	$0.0553280\pi$
$193$	4.80523i	0.345888i	−0.984932	+	0.172944i	$0.944672\pi$
$194$	14.9078	1.07032
$195$	17.5875i	1.25947i
$196$	4.68684	0.334775
$197$	−15.2225	−1.08456	−0.542280	−	0.840198i	$-0.682439\pi$
$197$	−15.2225	−1.08456	−0.542280	+	0.840198i	$0.682439\pi$
$198$	16.4599	1.16976
$199$	−12.1934	−0.864371	−0.432185	−	0.901785i	$-0.642257\pi$
$199$	−12.1934	−0.864371	−0.432185	+	0.901785i	$0.642257\pi$
$200$	− 2.39014i	− 0.169008i
$201$	8.47217i	0.597581i
$202$	−0.183919	−0.0129405
$203$	0	0
$204$	−5.80722	−0.406587
$205$	8.13019i	0.567837i
$206$	15.0468i	1.04836i
$207$	23.3551	1.62329
$208$	3.18562	0.220883
$209$	6.97723	0.482625
$210$	−8.39678	−0.579433
$211$	14.3852i	0.990318i	0.868802	+	0.495159i	$0.164890\pi$
$211$	14.3852i	0.990318i	−0.868802	+	0.495159i	$0.835110\pi$
$212$	−8.24904	−0.566546
$213$	20.7395i	1.42104i
$214$	0.304005i	0.0207814i
$215$	− 3.06387i	− 0.208954i
$216$	−19.4074	−1.32051
$217$	− 2.71085i	− 0.184024i
$218$	9.02387i	0.611174i
$219$	12.8090	0.865555
$220$	− 3.06387i	− 0.206566i
$221$	5.41328i	0.364137i
$222$	24.1621i	1.62166i
$223$	−12.7850	−0.856147	−0.428073	−	0.903744i	$-0.640808\pi$
$223$	−12.7850	−0.856147	−0.428073	+	0.903744i	$0.640808\pi$
$224$	1.52091i	0.101620i
$225$	20.7438	1.38292
$226$	8.09737	0.538629
$227$	26.4267	1.75400	0.877002	−	0.480487i	$-0.159540\pi$
$227$	26.4267	1.75400	0.877002	+	0.480487i	$0.159540\pi$
$228$	−12.5725	−0.832637
$229$	19.2020i	1.26890i	0.772963	+	0.634452i	$0.218774\pi$
$229$	19.2020i	1.26890i	−0.772963	+	0.634452i	$0.781226\pi$
$230$	− 4.34735i	− 0.286656i
$231$	9.85747	0.648574
$232$	0	0
$233$	−8.12368	−0.532200	−0.266100	−	0.963945i	$-0.585735\pi$
$233$	−8.12368	−0.532200	−0.266100	+	0.963945i	$0.585735\pi$
$234$	27.6478i	1.80739i
$235$	− 4.15288i	− 0.270904i
$236$	−2.95027	−0.192046
$237$	−31.5111	−2.04687
$238$	−2.58446	−0.167525
$239$	−30.3139	−1.96085	−0.980423	−	0.196903i	$-0.936912\pi$
$239$	−30.3139	−1.96085	−0.980423	+	0.196903i	$0.936912\pi$
$240$	5.52091i	0.356373i
$241$	−4.57387	−0.294629	−0.147315	−	0.989090i	$-0.547063\pi$
$241$	−4.57387	−0.294629	−0.147315	+	0.989090i	$0.547063\pi$
$242$	− 7.40314i	− 0.475892i
$243$	− 79.4564i	− 5.09713i
$244$	− 7.74593i	− 0.495883i
$245$	7.57163	0.483734
$246$	17.1986i	1.09654i
$247$	11.7197i	0.745705i
$248$	−1.78239	−0.113182
$249$	18.8750i	1.19615i
$250$	− 11.9388i	− 0.755077i
$251$	13.3162i	0.840511i	0.907406	+	0.420256i	$0.138060\pi$
$251$	13.3162i	0.840511i	−0.907406	+	0.420256i	$0.861940\pi$
$252$	−13.1998	−0.831511
$253$	5.10361i	0.320861i
$254$	9.51489	0.597017
$255$	−9.38161	−0.587499
$256$	1.00000	0.0625000
$257$	19.4778	1.21499	0.607495	−	0.794324i	$-0.292175\pi$
$257$	19.4778	1.21499	0.607495	+	0.794324i	$0.292175\pi$
$258$	− 6.48132i	− 0.403509i
$259$	10.7532i	0.668169i
$260$	5.14639	0.319166
$261$	0	0
$262$	14.2546	0.880649
$263$	− 10.0414i	− 0.619180i	−0.950870	−	0.309590i	$-0.899808\pi$
$263$	− 10.0414i	− 0.619180i	0.950870	−	0.309590i	$-0.100192\pi$
$264$	− 6.48132i	− 0.398897i
$265$	−13.3264	−0.818633
$266$	−5.59530	−0.343070
$267$	−54.5781	−3.34012
$268$	2.47909	0.151435
$269$	− 5.93600i	− 0.361924i	−0.983490	−	0.180962i	$-0.942079\pi$
$269$	− 5.93600i	− 0.361924i	0.983490	−	0.180962i	$-0.0579211\pi$
$270$	−31.3528	−1.90807
$271$	18.9788i	1.15288i	0.817140	+	0.576439i	$0.195558\pi$
$271$	18.9788i	1.15288i	−0.817140	+	0.576439i	$0.804442\pi$
$272$	1.69929i	0.103034i
$273$	16.5576i	1.00211i
$274$	−6.46454	−0.390537
$275$	4.53299i	0.273349i
$276$	− 9.19638i	− 0.553557i
$277$	−25.8106	−1.55081	−0.775405	−	0.631464i	$-0.782454\pi$
$277$	−25.8106	−1.55081	−0.775405	+	0.631464i	$0.782454\pi$
$278$	14.2573i	0.855098i
$279$	− 15.4692i	− 0.926119i
$280$	2.45703i	0.146836i
$281$	25.2296	1.50507	0.752537	−	0.658550i	$-0.228830\pi$
$281$	25.2296	1.50507	0.752537	+	0.658550i	$0.228830\pi$
$282$	− 8.78500i	− 0.523139i
$283$	−12.3473	−0.733974	−0.366987	−	0.930226i	$-0.619611\pi$
$283$	−12.3473	−0.733974	−0.366987	+	0.930226i	$0.619611\pi$
$284$	6.06870	0.360111
$285$	−20.3110	−1.20312
$286$	−6.04165	−0.357250
$287$	7.65410i	0.451807i
$288$	8.67893i	0.511411i
$289$	14.1124	0.830142
$290$	0	0
$291$	50.9467	2.98655
$292$	− 3.74813i	− 0.219343i
$293$	− 12.3492i	− 0.721447i	−0.932673	−	0.360724i	$-0.882530\pi$
$293$	− 12.3492i	− 0.721447i	0.932673	−	0.360724i	$-0.117470\pi$
$294$	16.0170	0.934132
$295$	−4.76618	−0.277498
$296$	7.07024	0.410949
$297$	36.8069	2.13575
$298$	− 8.23917i	− 0.477282i
$299$	−8.57254	−0.495763
$300$	− 8.16816i	− 0.471589i
$301$	− 2.88446i	− 0.166257i
$302$	− 22.0029i	− 1.26612i
$303$	−0.628533	−0.0361083
$304$	3.67893i	0.211001i
$305$	− 12.5136i	− 0.716527i
$306$	−14.7480	−0.843087
$307$	− 11.8371i	− 0.675579i	−0.941222	−	0.337790i	$-0.890321\pi$
$307$	− 11.8371i	− 0.675579i	0.941222	−	0.337790i	$-0.109679\pi$
$308$	− 2.88446i	− 0.164357i
$309$	51.4216i	2.92527i
$310$	−2.87946	−0.163543
$311$	10.1775i	0.577112i	0.957463	+	0.288556i	$0.0931751\pi$
$311$	10.1775i	0.577112i	−0.957463	+	0.288556i	$0.906825\pi$
$312$	10.8867	0.616337
$313$	−13.3421	−0.754142	−0.377071	−	0.926184i	$-0.623069\pi$
$313$	−13.3421	−0.754142	−0.377071	+	0.926184i	$0.623069\pi$
$314$	0.135585	0.00765148
$315$	−21.3244	−1.20150
$316$	9.22066i	0.518703i
$317$	− 11.9741i	− 0.672534i	−0.941767	−	0.336267i	$-0.890835\pi$
$317$	− 11.9741i	− 0.672534i	0.941767	−	0.336267i	$-0.109165\pi$
$318$	−28.1906	−1.58085
$319$	0	0
$320$	1.61551	0.0903096
$321$	1.03892i	0.0579870i
$322$	− 4.09277i	− 0.228081i
$323$	−6.25156	−0.347846
$324$	−40.2870	−2.23817
$325$	−7.61407	−0.422352
$326$	10.2536	0.567893
$327$	30.8386i	1.70538i
$328$	5.03259	0.277878
$329$	− 3.90969i	− 0.215548i
$330$	− 10.4706i	− 0.576388i
$331$	− 23.3952i	− 1.28592i	−0.765902	−	0.642958i	$-0.777707\pi$
$331$	− 23.3952i	− 1.28592i	0.765902	−	0.642958i	$-0.222293\pi$
$332$	5.52313	0.303121
$333$	61.3621i	3.36262i
$334$	16.7882i	0.918610i
$335$	4.00499	0.218816
$336$	5.19761i	0.283553i
$337$	22.8550i	1.24499i	0.782623	+	0.622496i	$0.213881\pi$
$337$	22.8550i	1.24499i	−0.782623	+	0.622496i	$0.786119\pi$
$338$	2.85183i	0.155119i
$339$	27.6723	1.50295
$340$	2.74521i	0.148880i
$341$	3.38037	0.183057
$342$	−31.9292	−1.72653
$343$	17.7746	0.959738
$344$	−1.89654	−0.102255
$345$	− 14.8568i	− 0.799864i
$346$	4.15473i	0.223360i
$347$	−6.48177	−0.347960	−0.173980	−	0.984749i	$-0.555663\pi$
$347$	−6.48177	−0.347960	−0.173980	+	0.984749i	$0.555663\pi$
$348$	0	0
$349$	−11.7835	−0.630754	−0.315377	−	0.948966i	$-0.602131\pi$
$349$	−11.7835	−0.630754	−0.315377	+	0.948966i	$0.602131\pi$
$350$	− 3.63517i	− 0.194308i
$351$	61.8247i	3.29996i
$352$	−1.89654	−0.101086
$353$	19.1188	1.01759	0.508795	−	0.860888i	$-0.330091\pi$
$353$	19.1188	1.01759	0.508795	+	0.860888i	$0.330091\pi$
$354$	−10.0824	−0.535873
$355$	9.80404	0.520344
$356$	15.9704i	0.846431i
$357$	−8.83224	−0.467452
$358$	10.6004i	0.560247i
$359$	8.29076i	0.437570i	0.975773	+	0.218785i	$0.0702093\pi$
$359$	8.29076i	0.437570i	−0.975773	+	0.218785i	$0.929791\pi$
$360$	14.0209i	0.738965i
$361$	5.46548	0.287657
$362$	− 21.7666i	− 1.14403i
$363$	− 25.2998i	− 1.32790i
$364$	4.84503	0.253948
$365$	− 6.05514i	− 0.316940i
$366$	− 26.4713i	− 1.38368i
$367$	− 28.8615i	− 1.50656i	−0.657700	−	0.753280i	$-0.728471\pi$
$367$	− 28.8615i	− 1.50656i	0.657700	−	0.753280i	$-0.271529\pi$
$368$	−2.69101	−0.140279
$369$	43.6775i	2.27376i
$370$	11.4220	0.593802
$371$	−12.5460	−0.651357
$372$	−6.09122	−0.315815
$373$	−22.3691	−1.15823	−0.579114	−	0.815247i	$-0.696601\pi$
$373$	−22.3691	−1.15823	−0.579114	+	0.815247i	$0.696601\pi$
$374$	− 3.22276i	− 0.166645i
$375$	− 40.8003i	− 2.10692i
$376$	−2.57063	−0.132570
$377$	0	0
$378$	−29.5169	−1.51818
$379$	14.8947i	0.765091i	0.923937	+	0.382545i	$0.124952\pi$
$379$	14.8947i	0.765091i	−0.923937	+	0.382545i	$0.875048\pi$
$380$	5.94334i	0.304887i
$381$	32.5166	1.66588
$382$	14.5209	0.742952
$383$	−10.9322	−0.558612	−0.279306	−	0.960202i	$-0.590104\pi$
$383$	−10.9322	−0.558612	−0.279306	+	0.960202i	$0.590104\pi$
$384$	3.41744	0.174396
$385$	− 4.65986i	− 0.237488i
$386$	4.80523	0.244580
$387$	− 16.4599i	− 0.836705i
$388$	− 14.9078i	− 0.756830i
$389$	15.6282i	0.792380i	0.918169	+	0.396190i	$0.129668\pi$
$389$	15.6282i	0.792380i	−0.918169	+	0.396190i	$0.870332\pi$
$390$	17.5875	0.890578
$391$	− 4.57280i	− 0.231256i
$392$	− 4.68684i	− 0.236721i
$393$	48.7142	2.45730
$394$	15.2225i	0.766899i
$395$	14.8960i	0.749501i
$396$	− 16.4599i	− 0.827142i
$397$	−7.66254	−0.384572	−0.192286	−	0.981339i	$-0.561590\pi$
$397$	−7.66254	−0.384572	−0.192286	+	0.981339i	$0.561590\pi$
$398$	12.1934i	0.611202i
$399$	−19.1216	−0.957280
$400$	−2.39014	−0.119507
$401$	4.71384	0.235398	0.117699	−	0.993049i	$-0.462448\pi$
$401$	4.71384	0.235398	0.117699	+	0.993049i	$0.462448\pi$
$402$	8.47217	0.422553
$403$	5.67802i	0.282842i
$404$	0.183919i	0.00915031i
$405$	−65.0840	−3.23405
$406$	0	0
$407$	−13.4090	−0.664658
$408$	5.80722i	0.287500i
$409$	7.70115i	0.380798i	0.981707	+	0.190399i	$0.0609781\pi$
$409$	7.70115i	0.380798i	−0.981707	+	0.190399i	$0.939022\pi$
$410$	8.13019	0.401521
$411$	−22.0922	−1.08973
$412$	15.0468	0.741303
$413$	−4.48709	−0.220795
$414$	− 23.3551i	− 1.14784i
$415$	8.92265	0.437996
$416$	− 3.18562i	− 0.156188i
$417$	48.7236i	2.38601i
$418$	− 6.97723i	− 0.341268i
$419$	25.7571	1.25832	0.629159	−	0.777277i	$-0.283400\pi$
$419$	25.7571	1.25832	0.629159	+	0.777277i	$0.283400\pi$
$420$	8.39678i	0.409721i
$421$	5.77170i	0.281296i	0.990060	+	0.140648i	$0.0449185\pi$
$421$	5.77170i	0.281296i	−0.990060	+	0.140648i	$0.955081\pi$
$422$	14.3852	0.700261
$423$	− 22.3104i	− 1.08477i
$424$	8.24904i	0.400609i
$425$	− 4.06153i	− 0.197013i
$426$	20.7395	1.00483
$427$	− 11.7808i	− 0.570115i
$428$	0.304005	0.0146947
$429$	−20.6470	−0.996847
$430$	−3.06387	−0.147753
$431$	−16.3606	−0.788063	−0.394032	−	0.919097i	$-0.628920\pi$
$431$	−16.3606	−0.788063	−0.394032	+	0.919097i	$0.628920\pi$
$432$	19.4074i	0.933740i
$433$	25.2730i	1.21454i	0.794494	+	0.607271i	$0.207736\pi$
$433$	25.2730i	1.21454i	−0.794494	+	0.607271i	$0.792264\pi$
$434$	−2.71085	−0.130125
$435$	0	0
$436$	9.02387	0.432165
$437$	− 9.90003i	− 0.473583i
$438$	− 12.8090i	− 0.612040i
$439$	−6.47777	−0.309167	−0.154583	−	0.987980i	$-0.549404\pi$
$439$	−6.47777	−0.309167	−0.154583	+	0.987980i	$0.549404\pi$
$440$	−3.06387	−0.146064
$441$	40.6768	1.93699
$442$	5.41328	0.257484
$443$	− 25.8813i	− 1.22966i	−0.788661	−	0.614828i	$-0.789225\pi$
$443$	− 25.8813i	− 1.22966i	0.788661	−	0.614828i	$-0.210775\pi$
$444$	24.1621	1.14668
$445$	25.8003i	1.22305i
$446$	12.7850i	0.605387i
$447$	− 28.1569i	− 1.33178i
$448$	1.52091	0.0718561
$449$	− 8.86533i	− 0.418381i	−0.977875	−	0.209190i	$-0.932917\pi$
$449$	− 8.86533i	− 0.418381i	0.977875	−	0.209190i	$-0.0670829\pi$
$450$	− 20.7438i	− 0.977873i
$451$	−9.54450	−0.449433
$452$	− 8.09737i	− 0.380868i
$453$	− 75.1937i	− 3.53291i
$454$	− 26.4267i	− 1.24027i
$455$	7.82718	0.366944
$456$	12.5725i	0.588763i
$457$	−22.1684	−1.03700	−0.518498	−	0.855079i	$-0.673509\pi$
$457$	−22.1684	−1.03700	−0.518498	+	0.855079i	$0.673509\pi$
$458$	19.2020	0.897250
$459$	−32.9788	−1.53932
$460$	−4.34735	−0.202696
$461$	38.5808i	1.79689i	0.439090	+	0.898443i	$0.355301\pi$
$461$	38.5808i	1.79689i	−0.439090	+	0.898443i	$0.644699\pi$
$462$	− 9.85747i	− 0.458611i
$463$	−30.1073	−1.39921	−0.699603	−	0.714531i	$-0.746640\pi$
$463$	−30.1073	−1.39921	−0.699603	+	0.714531i	$0.746640\pi$
$464$	0	0
$465$	−9.84041	−0.456338
$466$	8.12368i	0.376322i
$467$	− 5.64252i	− 0.261104i	−0.991441	−	0.130552i	$-0.958325\pi$
$467$	− 5.64252i	− 0.261104i	0.991441	−	0.130552i	$-0.0416750\pi$
$468$	27.6478	1.27802
$469$	3.77047	0.174104
$470$	−4.15288	−0.191558
$471$	0.463353	0.0213502
$472$	2.95027i	0.135797i
$473$	3.59686	0.165384
$474$	31.5111i	1.44735i
$475$	− 8.79314i	− 0.403457i
$476$	2.58446i	0.118458i
$477$	−71.5928	−3.27801
$478$	30.3139i	1.38653i
$479$	− 21.3948i	− 0.977554i	−0.872409	−	0.488777i	$-0.837443\pi$
$479$	− 21.3948i	− 0.977554i	0.872409	−	0.488777i	$-0.162557\pi$
$480$	5.52091	0.251994
$481$	− 22.5231i	− 1.02696i
$482$	4.57387i	0.208334i
$483$	− 13.9868i	− 0.636423i
$484$	−7.40314	−0.336506
$485$	− 24.0837i	− 1.09358i
$486$	−79.4564	−3.60421
$487$	12.6886	0.574977	0.287489	−	0.957784i	$-0.407180\pi$
$487$	12.6886	0.574977	0.287489	+	0.957784i	$0.407180\pi$
$488$	−7.74593	−0.350642
$489$	35.0410	1.58461
$490$	− 7.57163i	− 0.342051i
$491$	− 12.3249i	− 0.556214i	−0.960550	−	0.278107i	$-0.910293\pi$
$491$	− 12.3249i	− 0.556214i	0.960550	−	0.278107i	$-0.0897069\pi$
$492$	17.1986	0.775373
$493$	0	0
$494$	11.7197	0.527293
$495$	− 26.5911i	− 1.19518i
$496$	1.78239i	0.0800317i
$497$	9.22993	0.414019
$498$	18.8750	0.845808
$499$	3.86155	0.172867	0.0864335	−	0.996258i	$-0.472453\pi$
$499$	3.86155	0.172867	0.0864335	+	0.996258i	$0.472453\pi$
$500$	−11.9388	−0.533920
$501$	57.3728i	2.56323i
$502$	13.3162	0.594331
$503$	− 38.8953i	− 1.73426i	−0.498084	−	0.867129i	$-0.665963\pi$
$503$	− 38.8953i	− 1.73426i	0.498084	−	0.867129i	$-0.334037\pi$
$504$	13.1998i	0.587967i
$505$	0.297122i	0.0132218i
$506$	5.10361	0.226883
$507$	9.74596i	0.432833i
$508$	− 9.51489i	− 0.422155i
$509$	−32.3241	−1.43274	−0.716370	−	0.697721i	$-0.754198\pi$
$509$	−32.3241	−1.43274	−0.716370	+	0.697721i	$0.754198\pi$
$510$	9.38161i	0.415425i
$511$	− 5.70056i	− 0.252178i
$512$	− 1.00000i	− 0.0441942i
$513$	−71.3985	−3.15232
$514$	− 19.4778i	− 0.859127i
$515$	24.3082	1.07115
$516$	−6.48132	−0.285324
$517$	4.87531	0.214416
$518$	10.7532	0.472467
$519$	14.1986i	0.623248i
$520$	− 5.14639i	− 0.225684i
$521$	−7.79081	−0.341321	−0.170661	−	0.985330i	$-0.554590\pi$
$521$	−7.79081	−0.341321	−0.170661	+	0.985330i	$0.554590\pi$
$522$	0	0
$523$	−34.8524	−1.52399	−0.761995	−	0.647582i	$-0.775780\pi$
$523$	−34.8524	−1.52399	−0.761995	+	0.647582i	$0.775780\pi$
$524$	− 14.2546i	− 0.622713i
$525$	− 12.4230i	− 0.542184i
$526$	−10.0414	−0.437826
$527$	−3.02879	−0.131936
$528$	−6.48132	−0.282063
$529$	−15.7585	−0.685151
$530$	13.3264i	0.578861i
$531$	−25.6052	−1.11117
$532$	5.59530i	0.242587i
$533$	− 16.0319i	− 0.694420i
$534$	54.5781i	2.36182i
$535$	0.491123	0.0212331
$536$	− 2.47909i	− 0.107081i
$537$	36.2262i	1.56327i
$538$	−5.93600	−0.255919
$539$	8.88878i	0.382867i
$540$	31.3528i	1.34921i
$541$	− 12.2808i	− 0.527995i	−0.964523	−	0.263997i	$-0.914959\pi$
$541$	− 12.2808i	− 0.527995i	0.964523	−	0.263997i	$-0.0850410\pi$
$542$	18.9788	0.815208
$543$	− 74.3862i	− 3.19222i
$544$	1.69929	0.0728563
$545$	14.5781	0.624458
$546$	16.5576	0.708601
$547$	3.33790	0.142718	0.0713592	−	0.997451i	$-0.477266\pi$
$547$	3.33790	0.142718	0.0713592	+	0.997451i	$0.477266\pi$
$548$	6.46454i	0.276151i
$549$	− 67.2264i	− 2.86915i
$550$	4.53299	0.193287
$551$	0	0
$552$	−9.19638	−0.391424
$553$	14.0238i	0.596351i
$554$	25.8106i	1.09659i
$555$	39.0341	1.65691
$556$	14.2573	0.604645
$557$	−8.26501	−0.350200	−0.175100	−	0.984551i	$-0.556025\pi$
$557$	−8.26501	−0.350200	−0.175100	+	0.984551i	$0.556025\pi$
$558$	−15.4692	−0.654865
$559$	6.04165i	0.255535i
$560$	2.45703	0.103829
$561$	− 11.0136i	− 0.464995i
$562$	− 25.2296i	− 1.06425i
$563$	16.4004i	0.691193i	0.938383	+	0.345596i	$0.112323\pi$
$563$	16.4004i	0.691193i	−0.938383	+	0.345596i	$0.887677\pi$
$564$	−8.78500	−0.369915
$565$	− 13.0814i	− 0.550337i
$566$	12.3473i	0.518998i
$567$	−61.2728	−2.57321
$568$	− 6.06870i	− 0.254637i
$569$	− 18.6616i	− 0.782336i	−0.920319	−	0.391168i	$-0.872071\pi$
$569$	− 18.6616i	− 0.782336i	0.920319	−	0.391168i	$-0.127929\pi$
$570$	20.3110i	0.850735i
$571$	−39.1811	−1.63968	−0.819839	−	0.572594i	$-0.805937\pi$
$571$	−39.1811	−1.63968	−0.819839	+	0.572594i	$0.805937\pi$
$572$	6.04165i	0.252614i
$573$	49.6243	2.07308
$574$	7.65410	0.319476
$575$	6.43188	0.268228
$576$	8.67893	0.361622
$577$	7.81699i	0.325426i	0.986673	+	0.162713i	$0.0520244\pi$
$577$	7.81699i	0.325426i	−0.986673	+	0.162713i	$0.947976\pi$
$578$	− 14.1124i	− 0.586999i
$579$	16.4216	0.682459
$580$	0	0
$581$	8.40016	0.348497
$582$	− 50.9467i	− 2.11181i
$583$	− 15.6446i	− 0.647934i
$584$	−3.74813	−0.155099
$585$	44.6652	1.84668
$586$	−12.3492	−0.510140
$587$	−7.64544	−0.315561	−0.157780	−	0.987474i	$-0.550434\pi$
$587$	−7.64544	−0.315561	−0.157780	+	0.987474i	$0.550434\pi$
$588$	− 16.0170i	− 0.660531i
$589$	−6.55729	−0.270188
$590$	4.76618i	0.196221i
$591$	52.0221i	2.13990i
$592$	− 7.07024i	− 0.290585i
$593$	−9.75328	−0.400519	−0.200260	−	0.979743i	$-0.564179\pi$
$593$	−9.75328	−0.400519	−0.200260	+	0.979743i	$0.564179\pi$
$594$	− 36.8069i	− 1.51021i
$595$	4.17521i	0.171167i
$596$	−8.23917	−0.337490
$597$	41.6704i	1.70546i
$598$	8.57254i	0.350557i
$599$	21.4393i	0.875984i	0.898979	+	0.437992i	$0.144310\pi$
$599$	21.4393i	0.875984i	−0.898979	+	0.437992i	$0.855690\pi$
$600$	−8.16816	−0.333464
$601$	− 27.3326i	− 1.11492i	−0.830204	−	0.557459i	$-0.811776\pi$
$601$	− 27.3326i	− 1.11492i	0.830204	−	0.557459i	$-0.188224\pi$
$602$	−2.88446	−0.117562
$603$	21.5159	0.876194
$604$	−22.0029	−0.895285
$605$	−11.9598	−0.486236
$606$	0.628533i	0.0255324i
$607$	33.2949i	1.35140i	0.737177	+	0.675700i	$0.236158\pi$
$607$	33.2949i	1.35140i	−0.737177	+	0.675700i	$0.763842\pi$
$608$	3.67893	0.149200
$609$	0	0
$610$	−12.5136	−0.506661
$611$	8.18906i	0.331294i
$612$	14.7480i	0.596152i
$613$	−29.9946	−1.21147	−0.605736	−	0.795666i	$-0.707121\pi$
$613$	−29.9946	−1.21147	−0.605736	+	0.795666i	$0.707121\pi$
$614$	−11.8371	−0.477707
$615$	27.7845	1.12038
$616$	−2.88446	−0.116218
$617$	− 19.0269i	− 0.765992i	−0.923750	−	0.382996i	$-0.874892\pi$
$617$	− 19.0269i	− 0.765992i	0.923750	−	0.382996i	$-0.125108\pi$
$618$	51.4216	2.06848
$619$	− 41.0136i	− 1.64848i	−0.566243	−	0.824238i	$-0.691604\pi$
$619$	− 41.0136i	− 1.64848i	0.566243	−	0.824238i	$-0.308396\pi$
$620$	2.87946i	0.115642i
$621$	− 52.2256i	− 2.09574i
$622$	10.1775	0.408080
$623$	24.2895i	0.973139i
$624$	− 10.8867i	− 0.435816i
$625$	−7.33656	−0.293462
$626$	13.3421i	0.533259i
$627$	− 23.8443i	− 0.952250i
$628$	− 0.135585i	− 0.00541041i
$629$	12.0144	0.479044
$630$	21.3244i	0.849585i
$631$	−27.0196	−1.07563	−0.537816	−	0.843062i	$-0.680750\pi$
$631$	−27.0196	−1.07563	−0.537816	+	0.843062i	$0.680750\pi$
$632$	9.22066	0.366778
$633$	49.1606	1.95396
$634$	−11.9741	−0.475554
$635$	− 15.3714i	− 0.609994i
$636$	28.1906i	1.11783i
$637$	−14.9305	−0.591568
$638$	0	0
$639$	52.6699	2.08359
$640$	− 1.61551i	− 0.0638585i
$641$	− 30.9387i	− 1.22201i	−0.791629	−	0.611003i	$-0.790767\pi$
$641$	− 30.9387i	− 1.22201i	0.791629	−	0.611003i	$-0.209233\pi$
$642$	1.03892	0.0410030
$643$	2.10623	0.0830617	0.0415309	−	0.999137i	$-0.486777\pi$
$643$	2.10623	0.0830617	0.0415309	+	0.999137i	$0.486777\pi$
$644$	−4.09277	−0.161278
$645$	−10.4706	−0.412280
$646$	6.25156i	0.245964i
$647$	15.5641	0.611886	0.305943	−	0.952050i	$-0.401028\pi$
$647$	15.5641	0.611886	0.305943	+	0.952050i	$0.401028\pi$
$648$	40.2870i	1.58262i
$649$	− 5.59530i	− 0.219635i
$650$	7.61407i	0.298648i
$651$	−9.26417	−0.363092
$652$	− 10.2536i	− 0.401561i
$653$	− 44.9331i	− 1.75837i	−0.476481	−	0.879185i	$-0.658088\pi$
$653$	− 44.9331i	− 1.75837i	0.476481	−	0.879185i	$-0.341912\pi$
$654$	30.8386	1.20588
$655$	− 23.0283i	− 0.899792i
$656$	− 5.03259i	− 0.196490i
$657$	− 32.5298i	− 1.26911i
$658$	−3.90969	−0.152416
$659$	− 8.58905i	− 0.334582i	−0.985908	−	0.167291i	$-0.946498\pi$
$659$	− 8.58905i	− 0.334582i	0.985908	−	0.167291i	$-0.0535019\pi$
$660$	−10.4706	−0.407568
$661$	−3.16191	−0.122984	−0.0614921	−	0.998108i	$-0.519586\pi$
$661$	−3.16191	−0.122984	−0.0614921	+	0.998108i	$0.519586\pi$
$662$	−23.3952	−0.909280
$663$	18.4996	0.718465
$664$	− 5.52313i	− 0.214339i
$665$	9.03925i	0.350527i
$666$	61.3621	2.37773
$667$	0	0
$668$	16.7882	0.649555
$669$	43.6920i	1.68923i
$670$	− 4.00499i	− 0.154726i
$671$	14.6905	0.567119
$672$	5.19761	0.200502
$673$	−17.6361	−0.679821	−0.339911	−	0.940458i	$-0.610397\pi$
$673$	−17.6361	−0.679821	−0.339911	+	0.940458i	$0.610397\pi$
$674$	22.8550	0.880342
$675$	− 46.3864i	− 1.78541i
$676$	2.85183	0.109686
$677$	− 4.31076i	− 0.165676i	−0.996563	−	0.0828380i	$-0.973602\pi$
$677$	− 4.31076i	− 0.165676i	0.996563	−	0.0828380i	$-0.0263984\pi$
$678$	− 27.6723i	− 1.06275i
$679$	− 22.6734i	− 0.870125i
$680$	2.74521	0.105274
$681$	− 90.3119i	− 3.46076i
$682$	− 3.38037i	− 0.129441i
$683$	39.7629	1.52149	0.760743	−	0.649054i	$-0.224835\pi$
$683$	39.7629	1.52149	0.760743	+	0.649054i	$0.224835\pi$
$684$	31.9292i	1.22084i
$685$	10.4435i	0.399026i
$686$	− 17.7746i	− 0.678637i
$687$	65.6217	2.50363
$688$	1.89654i	0.0723049i
$689$	26.2783	1.00112
$690$	−14.8568	−0.565589
$691$	−18.4629	−0.702363	−0.351182	−	0.936307i	$-0.614220\pi$
$691$	−18.4629	−0.702363	−0.351182	+	0.936307i	$0.614220\pi$
$692$	4.15473	0.157939
$693$	− 25.0340i	− 0.950963i
$694$	6.48177i	0.246045i
$695$	23.0328	0.873684
$696$	0	0
$697$	8.55182	0.323923
$698$	11.7835i	0.446011i
$699$	27.7622i	1.05006i
$700$	−3.63517	−0.137397
$701$	−19.4613	−0.735042	−0.367521	−	0.930015i	$-0.619793\pi$
$701$	−19.4613	−0.735042	−0.367521	+	0.930015i	$0.619793\pi$
$702$	61.8247	2.33342
$703$	26.0109	0.981020
$704$	1.89654i	0.0714785i
$705$	−14.1922	−0.534510
$706$	− 19.1188i	− 0.719545i
$707$	0.279723i	0.0105201i
$708$	10.0824i	0.378919i
$709$	−6.61831	−0.248556	−0.124278	−	0.992247i	$-0.539661\pi$
$709$	−6.61831	−0.248556	−0.124278	+	0.992247i	$0.539661\pi$
$710$	− 9.80404i	− 0.367939i
$711$	80.0255i	3.00119i
$712$	15.9704	0.598517
$713$	− 4.79643i	− 0.179628i
$714$	8.83224i	0.330538i
$715$	9.76033i	0.365016i
$716$	10.6004	0.396154
$717$	103.596i	3.86887i
$718$	8.29076	0.309408
$719$	−18.1282	−0.676068	−0.338034	−	0.941134i	$-0.609762\pi$
$719$	−18.1282	−0.676068	−0.338034	+	0.941134i	$0.609762\pi$
$720$	14.0209	0.522527
$721$	22.8848	0.852274
$722$	− 5.46548i	− 0.203404i
$723$	15.6310i	0.581322i
$724$	−21.7666	−0.808949
$725$	0	0
$726$	−25.2998	−0.938965
$727$	14.2407i	0.528159i	0.964501	+	0.264079i	$0.0850681\pi$
$727$	14.2407i	0.528159i	−0.964501	+	0.264079i	$0.914932\pi$
$728$	− 4.84503i	− 0.179569i
$729$	−150.677	−5.58062
$730$	−6.05514	−0.224111
$731$	−3.22276	−0.119198
$732$	−26.4713	−0.978407
$733$	18.9693i	0.700648i	0.936629	+	0.350324i	$0.113928\pi$
$733$	18.9693i	0.700648i	−0.936629	+	0.350324i	$0.886072\pi$
$734$	−28.8615	−1.06530
$735$	− 25.8756i	− 0.954437i
$736$	2.69101i	0.0991920i
$737$	4.70170i	0.173189i
$738$	43.6775	1.60779
$739$	44.1573i	1.62435i	0.583412	+	0.812176i	$0.301717\pi$
$739$	44.1573i	1.62435i	−0.583412	+	0.812176i	$0.698283\pi$
$740$	− 11.4220i	− 0.419882i
$741$	40.0513	1.47132
$742$	12.5460i	0.460579i
$743$	32.3756i	1.18775i	0.804558	+	0.593874i	$0.202402\pi$
$743$	32.3756i	1.18775i	−0.804558	+	0.593874i	$0.797598\pi$
$744$	6.09122i	0.223315i
$745$	−13.3104	−0.487657
$746$	22.3691i	0.818990i
$747$	47.9348	1.75384
$748$	−3.22276	−0.117836
$749$	0.462364	0.0168944
$750$	−40.8003	−1.48981
$751$	− 27.0657i	− 0.987643i	−0.869563	−	0.493821i	$-0.835600\pi$
$751$	− 27.0657i	− 0.987643i	0.869563	−	0.493821i	$-0.164400\pi$
$752$	2.57063i	0.0937414i
$753$	45.5074	1.65838
$754$	0	0
$755$	−35.5458	−1.29364
$756$	29.5169i	1.07352i
$757$	49.9459i	1.81531i	0.419713	+	0.907657i	$0.362131\pi$
$757$	49.9459i	1.81531i	−0.419713	+	0.907657i	$0.637869\pi$
$758$	14.8947	0.541001
$759$	17.4413	0.633079
$760$	5.94334	0.215587
$761$	−8.75212	−0.317264	−0.158632	−	0.987338i	$-0.550708\pi$
$761$	−8.75212	−0.317264	−0.158632	+	0.987338i	$0.550708\pi$
$762$	− 32.5166i	− 1.17795i
$763$	13.7245	0.496859
$764$	− 14.5209i	− 0.525346i
$765$	23.8255i	0.861412i
$766$	10.9322i	0.394998i
$767$	9.39844	0.339358
$768$	− 3.41744i	− 0.123316i
$769$	22.0733i	0.795983i	0.917389	+	0.397992i	$0.130293\pi$
$769$	22.0733i	0.795983i	−0.917389	+	0.397992i	$0.869707\pi$
$770$	−4.65986	−0.167930
$771$	− 66.5642i	− 2.39725i
$772$	− 4.80523i	− 0.172944i
$773$	− 11.5345i	− 0.414865i	−0.978249	−	0.207433i	$-0.933489\pi$
$773$	− 11.5345i	− 0.414865i	0.978249	−	0.207433i	$-0.0665108\pi$
$774$	−16.4599	−0.591640
$775$	− 4.26016i	− 0.153029i
$776$	−14.9078	−0.535160
$777$	36.7484	1.31834
$778$	15.6282	0.560297
$779$	18.5145	0.663352
$780$	− 17.5875i	− 0.629734i
$781$	11.5095i	0.411844i
$782$	−4.57280	−0.163523
$783$	0	0
$784$	−4.68684	−0.167387
$785$	− 0.219038i	− 0.00781779i
$786$	− 48.7142i	− 1.73758i
$787$	−4.93112	−0.175775	−0.0878877	−	0.996130i	$-0.528012\pi$
$787$	−4.93112	−0.175775	−0.0878877	+	0.996130i	$0.528012\pi$
$788$	15.2225	0.542280
$789$	−34.3160	−1.22168
$790$	14.8960	0.529977
$791$	− 12.3153i	− 0.437883i
$792$	−16.4599	−0.584878
$793$	24.6756i	0.876256i
$794$	7.66254i	0.271933i
$795$	45.5422i	1.61521i
$796$	12.1934	0.432185
$797$	− 23.2713i	− 0.824312i	−0.911113	−	0.412156i	$-0.864776\pi$
$797$	− 23.2713i	− 0.824312i	0.911113	−	0.412156i	$-0.135224\pi$
$798$	19.1216i	0.676899i
$799$	−4.36825	−0.154537
$800$	2.39014i	0.0845041i
$801$	138.606i	4.89741i
$802$	− 4.71384i	− 0.166452i
$803$	7.10848	0.250853
$804$	− 8.47217i	− 0.298790i
$805$	−6.61190	−0.233039
$806$	5.67802	0.200000
$807$	−20.2860	−0.714099
$808$	0.183919	0.00647024
$809$	− 35.3445i	− 1.24265i	−0.783554	−	0.621324i	$-0.786595\pi$
$809$	− 35.3445i	− 1.24265i	0.783554	−	0.621324i	$-0.213405\pi$
$810$	65.0840i	2.28682i
$811$	9.47070	0.332561	0.166281	−	0.986078i	$-0.446824\pi$
$811$	9.47070	0.332561	0.166281	+	0.986078i	$0.446824\pi$
$812$	0	0
$813$	64.8589	2.27470
$814$	13.4090i	0.469984i
$815$	− 16.5647i	− 0.580237i
$816$	5.80722	0.203293
$817$	−6.97723	−0.244102
$818$	7.70115	0.269265
$819$	42.0497	1.46933
$820$	− 8.13019i	− 0.283918i
$821$	−29.4248	−1.02693	−0.513466	−	0.858110i	$-0.671639\pi$
$821$	−29.4248	−1.02693	−0.513466	+	0.858110i	$0.671639\pi$
$822$	22.0922i	0.770554i
$823$	− 29.4484i	− 1.02651i	−0.858238	−	0.513253i	$-0.828440\pi$
$823$	− 29.4484i	− 1.02651i	0.858238	−	0.513253i	$-0.171560\pi$
$824$	− 15.0468i	− 0.524180i
$825$	15.4912	0.539336
$826$	4.48709i	0.156126i
$827$	− 15.1182i	− 0.525710i	−0.964835	−	0.262855i	$-0.915336\pi$
$827$	− 15.1182i	− 0.525710i	0.964835	−	0.262855i	$-0.0846641\pi$
$828$	−23.3551	−0.811645
$829$	− 44.2208i	− 1.53585i	−0.640539	−	0.767925i	$-0.721289\pi$
$829$	− 44.2208i	− 1.53585i	0.640539	−	0.767925i	$-0.278711\pi$
$830$	− 8.92265i	− 0.309710i
$831$	88.2064i	3.05985i
$832$	−3.18562	−0.110441
$833$	− 7.96430i	− 0.275947i
$834$	48.7236	1.68716
$835$	27.1215	0.938577
$836$	−6.97723	−0.241313
$837$	−34.5916	−1.19566
$838$	− 25.7571i	− 0.889765i
$839$	4.77844i	0.164970i	0.996592	+	0.0824850i	$0.0262856\pi$
$839$	4.77844i	0.164970i	−0.996592	+	0.0824850i	$0.973714\pi$
$840$	8.39678	0.289716
$841$	0	0
$842$	5.77170	0.198906
$843$	− 86.2209i	− 2.96961i
$844$	− 14.3852i	− 0.495159i
$845$	4.60715	0.158491
$846$	−22.3104	−0.767046
$847$	−11.2595	−0.386880
$848$	8.24904	0.283273
$849$	42.1964i	1.44818i
$850$	−4.06153	−0.139309
$851$	19.0261i	0.652206i
$852$	− 20.7395i	− 0.710522i
$853$	7.44372i	0.254868i	0.991847	+	0.127434i	$0.0406741\pi$
$853$	7.44372i	0.254868i	−0.991847	+	0.127434i	$0.959326\pi$
$854$	−11.7808	−0.403132
$855$	51.5818i	1.76406i
$856$	− 0.304005i	− 0.0103907i
$857$	12.7075	0.434079	0.217039	−	0.976163i	$-0.430360\pi$
$857$	12.7075	0.434079	0.217039	+	0.976163i	$0.430360\pi$
$858$	20.6470i	0.704877i
$859$	17.2703i	0.589256i	0.955612	+	0.294628i	$0.0951958\pi$
$859$	17.2703i	0.589256i	−0.955612	+	0.294628i	$0.904804\pi$
$860$	3.06387i	0.104477i
$861$	26.1575	0.891444
$862$	16.3606i	0.557245i
$863$	36.4394	1.24041	0.620207	−	0.784439i	$-0.287049\pi$
$863$	36.4394	1.24041	0.620207	+	0.784439i	$0.287049\pi$
$864$	19.4074	0.660254
$865$	6.71200	0.228215
$866$	25.2730	0.858811
$867$	− 48.2284i	− 1.63792i
$868$	2.71085i	0.0920122i
$869$	−17.4873	−0.593217
$870$	0	0
$871$	−7.89745	−0.267595
$872$	− 9.02387i	− 0.305587i
$873$	− 129.384i	− 4.37898i
$874$	−9.90003	−0.334874
$875$	−18.1578	−0.613846
$876$	−12.8090	−0.432777
$877$	8.65427	0.292234	0.146117	−	0.989267i	$-0.453322\pi$
$877$	8.65427	0.292234	0.146117	+	0.989267i	$0.453322\pi$
$878$	6.47777i	0.218614i
$879$	−42.2027	−1.42346
$880$	3.06387i	0.103283i
$881$	− 7.63470i	− 0.257220i	−0.991695	−	0.128610i	$-0.958949\pi$
$881$	− 7.63470i	− 0.257220i	0.991695	−	0.128610i	$-0.0410515\pi$
$882$	− 40.6768i	− 1.36966i
$883$	0.617416	0.0207777	0.0103888	−	0.999946i	$-0.496693\pi$
$883$	0.617416	0.0207777	0.0103888	+	0.999946i	$0.496693\pi$
$884$	− 5.41328i	− 0.182068i
$885$	16.2882i	0.547521i
$886$	−25.8813	−0.869499
$887$	20.0952i	0.674732i	0.941374	+	0.337366i	$0.109536\pi$
$887$	20.0952i	0.674732i	−0.941374	+	0.337366i	$0.890464\pi$
$888$	− 24.1621i	− 0.810829i
$889$	− 14.4713i	− 0.485350i
$890$	25.8003	0.864830
$891$	− 76.4059i	− 2.55969i
$892$	12.7850	0.428073
$893$	−9.45718	−0.316472
$894$	−28.1569	−0.941708
$895$	17.1250	0.572424
$896$	− 1.52091i	− 0.0508099i
$897$	29.2962i	0.978171i
$898$	−8.86533	−0.295840
$899$	0	0
$900$	−20.7438	−0.691461
$901$	14.0175i	0.466990i
$902$	9.54450i	0.317797i
$903$	−9.85747	−0.328036
$904$	−8.09737	−0.269315
$905$	−35.1641	−1.16889
$906$	−75.1937	−2.49814
$907$	0.179212i	0.00595064i	0.999996	+	0.00297532i	$0.000947076\pi$
$907$	0.179212i	0.00595064i	−0.999996	+	0.00297532i	$0.999053\pi$
$908$	−26.4267	−0.877002
$909$	1.59622i	0.0529432i
$910$	− 7.82718i	− 0.259468i
$911$	43.3911i	1.43761i	0.695211	+	0.718806i	$0.255311\pi$
$911$	43.3911i	1.43761i	−0.695211	+	0.718806i	$0.744689\pi$
$912$	12.5725	0.416318
$913$	10.4748i	0.346666i
$914$	22.1684i	0.733267i
$915$	−42.7646	−1.41375
$916$	− 19.2020i	− 0.634452i
$917$	− 21.6798i	− 0.715931i
$918$	32.9788i	1.08846i
$919$	−18.8553	−0.621978	−0.310989	−	0.950413i	$-0.600660\pi$
$919$	−18.8553	−0.621978	−0.310989	+	0.950413i	$0.600660\pi$
$920$	4.34735i	0.143328i
$921$	−40.4526	−1.33296
$922$	38.5808	1.27059
$923$	−19.3326	−0.636340
$924$	−9.85747	−0.324287
$925$	16.8988i	0.555630i
$926$	30.1073i	0.989389i
$927$	130.590	4.28914
$928$	0	0
$929$	44.1780	1.44943	0.724717	−	0.689047i	$-0.241970\pi$
$929$	44.1780	1.44943	0.724717	+	0.689047i	$0.241970\pi$
$930$	9.84041i	0.322680i
$931$	− 17.2426i	− 0.565102i
$932$	8.12368	0.266100
$933$	34.7810	1.13868
$934$	−5.64252	−0.184629
$935$	−5.20640	−0.170267
$936$	− 27.6478i	− 0.903696i
$937$	−52.6808	−1.72101	−0.860503	−	0.509445i	$-0.829851\pi$
$937$	−52.6808	−1.72101	−0.860503	+	0.509445i	$0.829851\pi$
$938$	− 3.77047i	− 0.123110i
$939$	45.5960i	1.48797i
$940$	4.15288i	0.135452i
$941$	−21.9249	−0.714733	−0.357366	−	0.933964i	$-0.616325\pi$
$941$	−21.9249	−0.714733	−0.357366	+	0.933964i	$0.616325\pi$
$942$	− 0.463353i	− 0.0150968i
$943$	13.5428i	0.441013i
$944$	2.95027	0.0960232
$945$	47.6847i	1.55118i
$946$	− 3.59686i	− 0.116944i
$947$	− 6.64514i	− 0.215938i	−0.994154	−	0.107969i	$-0.965565\pi$
$947$	− 6.64514i	− 0.215938i	0.994154	−	0.107969i	$-0.0344347\pi$
$948$	31.5111	1.02343
$949$	11.9401i	0.387593i
$950$	−8.79314	−0.285287
$951$	−40.9209	−1.32695
$952$	2.58446	0.0837627
$953$	2.98044	0.0965461	0.0482730	−	0.998834i	$-0.484628\pi$
$953$	2.98044	0.0965461	0.0482730	+	0.998834i	$0.484628\pi$
$954$	71.5928i	2.31790i
$955$	− 23.4586i	− 0.759101i
$956$	30.3139	0.980423
$957$	0	0
$958$	−21.3948	−0.691235
$959$	9.83196i	0.317490i
$960$	− 5.52091i	− 0.178186i
$961$	27.8231	0.897519
$962$	−22.5231	−0.726174
$963$	2.63844	0.0850226
$964$	4.57387	0.147315
$965$	− 7.76289i	− 0.249896i
$966$	−13.9868	−0.450019
$967$	6.32629i	0.203440i	0.994813	+	0.101720i	$0.0324346\pi$
$967$	6.32629i	0.203440i	−0.994813	+	0.101720i	$0.967565\pi$
$968$	7.40314i	0.237946i
$969$	21.3644i	0.686322i
$970$	−24.0837	−0.773281
$971$	− 20.1311i	− 0.646036i	−0.946393	−	0.323018i	$-0.895303\pi$
$971$	− 20.1311i	− 0.646036i	0.946393	−	0.323018i	$-0.104697\pi$
$972$	79.4564i	2.54856i
$973$	21.6841	0.695159
$974$	− 12.6886i	− 0.406570i
$975$	26.0207i	0.833328i
$976$	7.74593i	0.247941i
$977$	−27.6748	−0.885394	−0.442697	−	0.896671i	$-0.645978\pi$
$977$	−27.6748	−0.885394	−0.442697	+	0.896671i	$0.645978\pi$
$978$	− 35.0410i	− 1.12049i
$979$	−30.2885	−0.968026
$980$	−7.57163	−0.241867
$981$	78.3175	2.50049
$982$	−12.3249	−0.393303
$983$	− 5.92730i	− 0.189052i	−0.995522	−	0.0945258i	$-0.969867\pi$
$983$	− 5.92730i	− 0.189052i	0.995522	−	0.0945258i	$-0.0301335\pi$
$984$	− 17.1986i	− 0.548271i
$985$	24.5921	0.783569
$986$	0	0
$987$	−13.3612	−0.425290
$988$	− 11.7197i	− 0.372852i
$989$	− 5.10361i	− 0.162285i
$990$	−26.5911	−0.845121
$991$	−61.6302	−1.95775	−0.978874	−	0.204464i	$-0.934455\pi$
$991$	−61.6302	−1.95775	−0.978874	+	0.204464i	$0.934455\pi$
$992$	1.78239	0.0565909
$993$	−79.9517	−2.53719
$994$	− 9.22993i	− 0.292756i
$995$	19.6986	0.624488
$996$	− 18.8750i	− 0.598077i
$997$	− 48.6851i	− 1.54187i	−0.636911	−	0.770937i	$-0.719788\pi$
$997$	− 48.6851i	− 1.54187i	0.636911	−	0.770937i	$-0.280212\pi$
$998$	− 3.86155i	− 0.122235i
$999$	137.215	4.34129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.b.i.1681.1		12
29.8	odd	28		58.2.d.b.23.1	✓	12
29.12	odd	4		1682.2.a.t.1.6		6
29.17	odd	4		1682.2.a.q.1.1		6
29.18	odd	28		58.2.d.b.53.1	yes	12
29.28	even	2	inner	1682.2.b.i.1681.12		12
87.8	even	28		522.2.k.h.487.1		12
87.47	even	28		522.2.k.h.343.1		12
116.47	even	28		464.2.u.h.401.2		12
116.95	even	28		464.2.u.h.81.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.d.b.23.1	✓	12	29.8	odd	28
58.2.d.b.53.1	yes	12	29.18	odd	28
464.2.u.h.81.2		12	116.95	even	28
464.2.u.h.401.2		12	116.47	even	28
522.2.k.h.343.1		12	87.47	even	28
522.2.k.h.487.1		12	87.8	even	28
1682.2.a.q.1.1		6	29.17	odd	4
1682.2.a.t.1.6		6	29.12	odd	4
1682.2.b.i.1681.1		12	1.1	even	1	trivial
1682.2.b.i.1681.12		12	29.28	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -3.41744i q^{3} -1.00000 q^{4} -1.61551 q^{5} -3.41744 q^{6} -1.52091 q^{7} +1.00000i q^{8} -8.67893 q^{9} +1.61551i q^{10} -1.89654i q^{11} +3.41744i q^{12} +3.18562 q^{13} +1.52091i q^{14} +5.52091i q^{15} +1.00000 q^{16} +1.69929i q^{17} +8.67893i q^{18} +3.67893i q^{19} +1.61551 q^{20} +5.19761i q^{21} -1.89654 q^{22} -2.69101 q^{23} +3.41744 q^{24} -2.39014 q^{25} -3.18562i q^{26} +19.4074i q^{27} +1.52091 q^{28} +5.52091 q^{30} +1.78239i q^{31} -1.00000i q^{32} -6.48132 q^{33} +1.69929 q^{34} +2.45703 q^{35} +8.67893 q^{36} -7.07024i q^{37} +3.67893 q^{38} -10.8867i q^{39} -1.61551i q^{40} -5.03259i q^{41} +5.19761 q^{42} +1.89654i q^{43} +1.89654i q^{44} +14.0209 q^{45} +2.69101i q^{46} +2.57063i q^{47} -3.41744i q^{48} -4.68684 q^{49} +2.39014i q^{50} +5.80722 q^{51} -3.18562 q^{52} +8.24904 q^{53} +19.4074 q^{54} +3.06387i q^{55} -1.52091i q^{56} +12.5725 q^{57} +2.95027 q^{59} -5.52091i q^{60} +7.74593i q^{61} +1.78239 q^{62} +13.1998 q^{63} -1.00000 q^{64} -5.14639 q^{65} +6.48132i q^{66} -2.47909 q^{67} -1.69929i q^{68} +9.19638i q^{69} -2.45703i q^{70} -6.06870 q^{71} -8.67893i q^{72} +3.74813i q^{73} -7.07024 q^{74} +8.16816i q^{75} -3.67893i q^{76} +2.88446i q^{77} -10.8867 q^{78} -9.22066i q^{79} -1.61551 q^{80} +40.2870 q^{81} -5.03259 q^{82} -5.52313 q^{83} -5.19761i q^{84} -2.74521i q^{85} +1.89654 q^{86} +1.89654 q^{88} -15.9704i q^{89} -14.0209i q^{90} -4.84503 q^{91} +2.69101 q^{92} +6.09122 q^{93} +2.57063 q^{94} -5.94334i q^{95} -3.41744 q^{96} +14.9078i q^{97} +4.68684i q^{98} +16.4599i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} - 4 q^{6} - 6 q^{7} - 24 q^{9} + 6 q^{13} + 12 q^{16} + 2 q^{22} - 28 q^{23} + 4 q^{24} + 8 q^{25} + 6 q^{28} + 54 q^{30} - 40 q^{33} + 12 q^{34} + 18 q^{35} + 24 q^{36} - 36 q^{38} + 20 q^{42}+ \cdots - 4 q^{96}+O(q^{100}) \) 12 * q - 12 * q^4 - 4 * q^6 - 6 * q^7 - 24 * q^9 + 6 * q^13 + 12 * q^16 + 2 * q^22 - 28 * q^23 + 4 * q^24 + 8 * q^25 + 6 * q^28 + 54 * q^30 - 40 * q^33 + 12 * q^34 + 18 * q^35 + 24 * q^36 - 36 * q^38 + 20 * q^42 + 36 * q^45 + 2 * q^49 + 22 * q^51 - 6 * q^52 + 6 * q^53 + 46 * q^54 + 38 * q^57 + 38 * q^59 - 34 * q^62 + 66 * q^63 - 12 * q^64 + 22 * q^65 - 42 * q^67 + 18 * q^71 + 24 * q^74 + 20 * q^78 + 4 * q^81 - 30 * q^82 - 4 * q^83 - 2 * q^86 - 2 * q^88 + 44 * q^91 + 28 * q^92 - 2 * q^93 + 16 * q^94 - 4 * q^96

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